A194377 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(6) and < > denotes fractional part.

1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Author

Clark Kimberling, Aug 23 2011

Keywords

• nonn

Comments

See A194368. Although a(n)=A007957(n) for n = 1..70, the number 208, for example, is here but not A007957.

Crossrefs

Cf. A007957, A194368, A194376.

Programs

Mathematica

r = Sqrt[6]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 500}];
Flatten[Position[t1, 1]]   (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t2, 1]]   (* A194376 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]   (* A194377 *)

PARI

is(n)=my(r=sqrt(6),f=x->x-x\1);sum(k=1,n,f(1/2+k*r)-f(k*r))>0 \\ Charles R Greathouse IV, Jul 25 2012